Minimum permanents of doubly stochastic matrices with One fixed entry

Abstract

For n  3 and sufficiently small a  0, the minimum value of the permanent function restricted on n × n doubly stochastic matrices with at least one entry equal to a is obtained. For n = 3, the explicit form of the function p is derived where p(a) = min{per(C):C = (c_ij )∊Ω₃ c ₁₁ = a}a∊ [0, 1].